Kutta-Joukowsky Condition Research Articles

Numerical Analysis of Leading-Edge Vortex Effect on Tidal Current Energy Extraction Performance for Chord-Wise Deformable Oscillating Hydrofoil

To improve the energy extraction performance of the oscillating hydrofoil, the lift force that acts on the oscillating hydrofoil is analyzed. The pressure difference between the oscillating hydrofoil‘s opposing surfaces is dominant to generate the lift force. Forming and shedding of the leading-edge vortex from the hydrofoil surface determines the pressure difference between the opposing surfaces of the oscillating hydrofoil. In this paper, the hydrofoil with different chord flexibility coefficients and maximum offset at the trailing edge are analyzed to obtain the power coefficient, lift coefficient, and moment coefficient of the oscillating hydrofoil. The influence mechanism of chord-wise deformation of the oscillating hydrofoil on the energy extraction performance is explored. According to the Kutta–Joukowsky condition and the Stokes’ theorem, the relationship between the attached vortex on the hydrofoil and the surface pressure of the hydrofoil, the surface pressure difference of the hydrofoil, and the lift force that acts on the hydrofoil are investigated. By quantifying the vortex intensity, the ascending-shedding process of the attached vortex on the hydrofoil is characterized. Finally, the complete influence chain among the chord-wise flexure, the attached vortex on the hydrofoil, and the energy extraction performance of the oscillating hydrofoil is established.

The performance of a centrifugal compressor with a tandem impeller in off-design conditions

Centrifugal compressors often suffer relatively low efficiency and a terrible operating range particularly due to the complex flow structure and intense impeller/diffuser interaction. Numerous studies have focused on improving the centrifugal compressor performance using many innovative ideas, such as the tandem impeller, which has become increasingly attractive due to its ability to achieve the flow control with no additional air supply configurations and control costs in compressor. However, few studies that attempted to the investigation of tandem impeller have been published until now and the results are always contradictory. To explore the potential of the tandem impeller to enhance the compressor performance and the underlying mechanism of the flow phenomena in the tandem impellers, this paper numerically investigated a high-pressure-ratio centrifugal compressor with several tandem impellers at off-design operating speeds. The results encouragingly demonstrate that the tandem impeller can achieve a performance enhancement over a wide range of operating conditions. Approximately 1.8% maximum enhancement in isentropic efficiency and 5.0% maximum enhancement in operating range are achieved with the inducer/exducer circumferential displacement of [Formula: see text] = 25% and 50%, respectively. The observed stage performance gain of the tandem impellers decreases when the operating speed increases due to the increased inducer shock, increased wake losses, and deteriorated tandem impeller discharge flow uniformity. In addition, the tandem impeller can extend the impeller operating range particularly at low rotation speeds, which is found to be a result from the suppression of the low-momentum fluid radial movement. The results also indicate that the maximum flux capacity of the tandem impeller decreases due to the restriction of the inducer airfoil Kutta–Joukowsky condition.

Origin of ejecta in the water impact problem

This work presents the analysis of the flow resulting from the flat plate impact on the surface of an incompressible viscous liquid at zero deadrise angle—of particular interest is the flow structure near the plate edge, r → 0, evolving at early times, t → 0. The deduced mathematical formulation proves to be of a singular perturbation type with the underlying governing equations having a linear structure. The key goals here are to elucidate the effects of viscosity and surface tension, which turn out to contribute to the solution at the leading order, and to resolve both t → 0 and r → 0 limit singularities in the classical pressure-impulse theory. In the course of construction of the solution, first the standard assumptions behind the existence of the inviscid approximation are revisited, which leads to correcting the previously known interpretation of the self-similarity of the classical inviscid solution near the plate edge. Second, new scalings of the solution structure near the plate edge are determined, with which the viscous solution near the edge is constructed analytically and matched to the inviscid one. Finally, the analysis of both the Stokes and inviscid limits of this uniformly valid solution allows us to uncover the scalings for the early time-evolution of ejecta—a jet forming during the impact—as well as to clarify the applicability of the Kutta-Joukowsky condition used in previous studies.

Vortical structures in the wake of the savonius wind turbine by the discrete vortex method

This paper treats the vortex shedding phenomenon of a savonius wind turbine, whose knowledge is primordial in correctly calculating the airloads on the blades. The specific aim being to numerically predict the disposition and geometry of the vortical structures in the wake of the savonius rotor whose existence has been visualised by a number of experimentalists. In the numerical approach, the blade is represented by discrete bound vortices while the wake is generated in a time stepping calculation as an emission of free vortices. The calculations are enhanced by the Newmann boundary condition coupled to the Kutta–Joukowsky condition and the Kelvin's theorem for the conservation of circulation. The convection of the vortices in the wake is accomplished through a predictor corrector integration scheme. A code has been developed which predicts the wake structure to be in good agreement with the experimental visualizations: For low tip speed ratios, the wake consists of a series of three discrete vortical structures while at higher tip speed ratios, the characteristic structure is the presences of a central vortex.

An immersed interface solver for the 2-D unbounded Poisson equation and its application to potential flow

This paper presents a novel algorithm to solve the 2-D potential flow past complex geometries with circulation in unbounded domain and in the presence of a given vorticity field. It is based on a Poisson solver that combines two components: the immersed interface method to enforce the boundary condition on each inner boundary and the James–Lackner algorithm to compute the outer boundary condition consistent with the unbounded domain solution. The algorithm is here based on second order finite differences and it requires solely 1-D stencil corrections; this makes the immersed interface part of the present method easily extendable to higher dimensional problems. The treatment of the outer boundaries requires an iterative boundary potential method. The algorithm is validated, by means of grid convergence studies, on the flow past multiple bodies. The results confirm the second order accuracy everywhere. The algorithm is also self consistent as “all is done on the grid” (thus without using a vortex panel boundary element method in addition to the grid). For cusped airfoils, a consistent way to enforce the Kutta–Joukowsky condition is also presented. The present algorithm constitutes a crucial building block towards an immersed interface-enabled vortex particle-mesh method for the computation of unsteady viscous flows, with boundary layers, detached shear layers and wakes. A possible extension to 3-D problems is also briefly discussed.

The unsteady pressure field and the aerodynamic performances of a Savonius rotor based on the discrete vortex method

The aim of this paper is to numerically explore the non-linear two-dimensional unsteady potential flow over a Savonius rotor and to develop a code for predicting its aerodynamics performances. In the model developed, the rotor is represented in a median plane by two semicircles, displaced along their common diameter. The two semicircles can be considered to produce lifting effects. As a result, they are modelled by a collection of discrete vortices on their contours. The flow field is then governed by the Laplace equation. The versatile Neumann boundary condition, applied over the contour of the semicircles and the Kutta Joukowsky condition applied at the four extremities of the semicircles have been used in the modelling. The torque distribution of the stationary rotor and the unsteady pressure field on the blades of the rotating rotor, predicted by the code developed, have been compared and validated by some experimental data.

Matrix Solution of Compressible Flow on S1 Surface Through a Turbomachine Blade Row With Splitter Vanes or Tandem Blades

The basic aerothermodynamic equations of turbomachine flow expressed with respect to nonorthogonal curvilinear coordinates and corresponding nonorthogonal velocity components are used to solve the compressible flow S1 surface in a turbomachine blade row with splitter vanes or tandem blades. The equation of stream function is solved by matrix technique, and the mass flow ratio and outlet flow angle are determined by applying the Kutta-Joukowsky condition to the trailing edges of the main blade and the splitter vane. Typical examples are given to illustrate the effectiveness of the present method.

Trailing Edge Conditions for Unsteady Flows at High Reduced Frequency

In the prediction of time-variant flows past isolated airfoils and airfoil cascades, the steady-state KuttaJoukowsky condition is generally assumed to be valid. Recent experimental investigations on isolated airfoils however, have indicated that this assumption is only appropriate for flows characterized by low reduced frequency values. For the case of airfoil cascades, only low reduced frequency data exist, but these limited data indicate the same trend, i.e., the Kutta-Joukowsky condition is appropriate only for low reduced frequency values. Hence, the objective of the experimental investigation described herein was to determine the applicability of the Kutta-Joukowsky condition for isolated airfoils and airfoil cascades at high reduced frequency values over a range of incidence angles. This was accomplished by measuring the time-variant surface pressure distribution in the trailing edge region of a classical isolated flat plate, a classical flat plate cascade, and a cambered airfoil cascade, and correlating these data with an appropriate state-of-the-art zero incidence flat plate theoretical model that applies the Kutta-Joukowsky condition. The results obtained indicate that at these high reduced frequency values, the Kutta-Joukowsky condition is appropriate for the classical isolated flat plate and flat plate cascade over a wide range of incidence angles. However, for the cambered airfoil cascade, this condition does not appear to be satisfied for any value of incidence angle.

Experimental Studies of Unsteady Trailing-Edge Conditions

In the prediction of unsteady pressure distributions over airfoils, the steady-state Kutta-Joukowsky condition usually is assumed. Recent experimental investigations show that the pressure differential at the trailing edge approaches zero at lower reduced frequencies (ft 5). This paper describes an investigation to find the range of reduced frequencies over which the classical Kutta-Joukowsky condition is valid and the nature of deviations beyond this range. An NACA 64A010 airfoil with a 15-cm chord is supported between the side wails of a 25- x35-cm low-speed wind tunnel and oscillated over reduced frequencies ranging from 0.05 to 1.2. The experimentally determined loading near the trailing edge is measured and compared with unsteady, incompressible, small-disturbance theory. Experimental data are obtained both with and without a boundary-layer trip. It is observed that application of the Kutta-Joukowsky condition in the theoretical analysis is valid below a reduced frequency of about 0.6 in predicting the loading in the trailing-edge region. At reduced frequencies beyond 0.8, the predicted results underestimate the measured loading near the trailing edge, and the deviations increase further with reduced frequency. Similarly, the experimental phase angle of the loading in the trailing-edge region agrees reasonably with the linear theory up to k = 0.8 but lags the predicted value beyond k - 0.8.

REFRACTION ANGLES FOR LUMINOUS SOURCES WITHIN THE ATMOSPHERE

1 Sears, W. R. and Resler, E. L., Theory of thin airfoils in fluids of high electrical conductivity, J. Fluid Mech. 5, 257 (1959); also McCune, J. E. and Resler, E. L., Compressibility effects in magnetoaerodynamic flows past thin bodies, J. Aerospace Sci. 27, 493 (1960). 2 Grad, H., Reducible problems in magneto-fluid dynamic steady flows, Revs. Mod. Phys. 32, 830 (1960). 3 Cumberbatch, E., Sarason, L., and Weitzner, H., A magnetofluid-dynamic Kutta-Joukowsky condition, J. Aerospace Sci. 29, 244 (1962). 4 Mikhlin, S. G., Integral Equations (Pergamon Press, New York, 1957), p. 126. Stewartson, K., Magneto-fluid dynamics of thin bodies in oblique fields, Z. Angew. Math. Phys. 12, 261 (1961).